Bayesian Population PharmacokineticPharmacodynamic Modeling

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Bayesian Population Pharmacokinetic/Pharmacodynami
c Modeling

• Steven Kathman
• GlaxoSmithKline

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Half of the modern drugs could well be thrown out
of the window, except that the birds might eat
them. Dr. Martin Henry Fischer
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Outline

• Introduction
• Population PK modeling
• Population PK/PD modeling
• Modeling the time course of ANC
• Other examples
• Conclusions

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Introduction

• KSP inhibitor (Ispinesib) being developed for the
  treatment of cancer.
• Blocks assembly of a functional mitotic spindle
  and leads to G2/M arrest.
• Causes cell cycle arrest in mitosis and
  subsequent cell death.
• Leads to a transient reduction in absolute
  neutrophil counts (ANC).

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Introduction

• KSP10001 was the FTIH study.
• Ispinesib dosed once every three weeks.
• PK data collected after first dose.
• ANC assessed on Days 1 (pre-dose), 8, 15, and 22
  (C2D1 pre-dose). More frequent assessments done
  if ANC lt 0.75 (109/L).
• Prolonged Grade 4 neutropenia (gt 5 days) most
  common DLT.

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Objectives

• Determine a suitable PK model.
• - Examine 2 vs 3 compartment models.
• Determine a suitable model for PD endpoint (i.e.,
  time course of absolute neutrophil counts).
• - Using Nonlinear mixed models.
• - Using Bayesian methods.

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Pharmacokinetics

• The action of drugs in the body over a period of
  time, including the processes of absorption,
  distribution, localisation in tissues,
  biotransformation and excretion.
• Simple terms what happens to the drug after it
  enters the body.
• What is the body doing to the drug over time?

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R
A2 C2V2
A1 C1V1
k12
k21
k10
dA1/dt R k21A2 k12A1 k10A1
dA2/dt k12A1 k21A2
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CL k10V1 Q k12V1 k21V2
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Infusion
k0 zero order infusion rate Tt during
infusion, constant time infusion was stopped
after infusion.
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PK Model
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PK Model
µVague MVN prior
R chosen based on CV30
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If that was painful
In mathematics you don't understand things. You
just get used to them. Johann von Neumann (1903
- 1957)
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Bayesian Results

• Typical Bayesian analysis (via MCMC) involves
  estimation of the joint posterior distribution of
  all unobserved stochastic quantities conditional
  on observed data.
• Generating random samples from the joint
  posterior distribution of the parameters.
• Marginal distribution of each parameter is
  completely characterized (numerical integration).

P(individual specific PK parameters, population
PK parameters PK data)
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R
k12
k13
A1C1V1
A2C2V2
A3C3V3
k31
k21
k10
dA1/dt R k21A2 k31A3 k12 A1 k13A1 k10
A1
dA2/dt k12A1 k21A2
dA3/dt k13A1 k31A3
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Pharmacodynamics

• The study of the biochemical and physiological
  effects of drugs and the mechanisms of their
  actions, including the correlation of actions and
  effects of drugs with their chemical structure,
  also, such effects on the actions of a particular
  drug or drugs.
• What is the drug doing to the body?

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Modeling the Time Course Absolute Neutrophil
Counts
When you are curious, you find lots of
interesting things to do. The way to get started
is to quit talking and begin doing. Walt
Disney (1901-1966)
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Model of Myelosuppression
Prol
Circ
Transit 1
Transit 2
Transit 3
ktr
ktr
ktr
ktr
kprol ktr
kcirc ktr
EDrug ß?Conc
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Features of Model

• Proliferating compartment sensitive to drug.
• Three transit compartments represent
  maturation.
• Compartment of circulating blood cells.
• System parameters MTT, baseline, and feedback.
• Drug specific parameter Slope.

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Feedback

• Account for rebound phase (overshoot).
• Negative feedback from circulating cells to
  proliferative cells.
• G-CSF levels increase when circulating neutrophil
  counts are low.
• G-CSF stimulates proliferation in bone marrow.

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Model of Myelosuppression

• dProl/dt kprolProl(1-EDrug)(Circ0/Circ)?-ktr
  Prol
• dTransit1/dt ktrProl-ktrTransit1
• dTransit2/dt ktrTransit1-ktrTransit2
• dTransit3/dt ktrTransit2-ktrTransit3
• dCirc/dt ktrTransit3-kcircCirc

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ANCijt(Meanij(MTTi, Circ0(i),?, ßi Concij),
?ij, 4)
Mean Solution of the differential equation
(Circ) MTTi 4/(ktr(i)) Mean transit time.
Fairly informative priors (Literature).
ln(MTTi)N(?MTT, ?MTT) ln(Circ0(i))N(?circ,
?circ) ln(ßi)N(?ß, ?ß)
Vague prior.
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Simulate New Schedule

• Using mechanistic/semi-physiological models
  allows for simulation of new schedules.
• Simulate dosing on days 1, 8, and 15 repeated
  every 28 days.
• PK/PD model accurately predicted the observed
  severity and duration of neutropenia.

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Why Bayesian?

• Incorporate prior information (MTT and baseline).
• Better integration algorithm (Monte Carlo vs
  Taylor Series or Quadrature).
• Posterior distribution vs MLE More informative,
  avoids potentially problematic maximization
  algorithms.
• Better individual estimates Bayesian vs
  Empirical Bayesian (which usually fail to account
  for estimated population parameters?).

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Tumor Growth Models

• dC/dt KLC(t) KDC(t)D(t)exp(-?t)
• where KL Tumor growth rate
• KD Drug constant kill rate
• D(t) Dose or PK measure
• ? rate constant for resistance
• dC/dt exp(?1t) C(t) KDC(t)D(t)exp(-?2t)

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Preclinical PK

• Concentrations in plasma.
• Concentrations in a tumor.
• Relate the two
• Plasma two-compartment model.
• Tumor
• dCT(t)/dt (KP/VT)AP(t)-KTCT(t)

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More PK

• Compound given through iv infusion.
• Should be 1-hr infusion.
• Reason to believe that the infusion time is less
  for some subjects.
• Making the infusion times a parameter to be
  estimated, with informative priors.

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Software

• WinBugs (Pharmaco and WBDiff)
• - Pharmaco Built in PK functions.
• - WBDiff Differential Equation Solver
• NONMEM
• SAS macro
• R nlmeODE library and function

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Conclusions

• PK/PD modeling often involves interesting and
  complicated models.
• Models can serve many useful functions in drug
  development.
• Bayesian methods help with
• Better algorithms
• More flexibility
• Incorporating outside information

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General Remarks

• PK/PD modeling involves different skills coming
  together (medical, pharmacokinetics,
  pharmacology, statistics, etc.).
• As a statistician, helps to develop knowledge in
  areas outside of statistics.

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References
Knowledge is of two kinds. We know a subject
ourselves, or we know where we can find
information on it. Samuel Johnson (1709 - 1784),
quoted in Boswell's Life of Johnson
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References

• Gibaldi, M. and Perrier, D. (1982)
  Pharmacokinetics.
• Friberg, L. et. al. (2002). Model of
  Chemotherapy-Induced Myelosuppression with
  Parameter Consistency Across Drugs. JCO
  204713-4721.
• Friberg, L. et. al. (2003). Mechanistic Models
  for Myelosuppression. Investigational New Drugs
  21183-194.
• Lunn, D. et. al. (2002). Bayesian Analysis of
  Population PK/PD Models General Concepts and
  Software. Journal of PK and PD 29271-307.
• PK Bugs User Guide.
• Christian, R. and Casella, G. (2005) Monte Carlo
  Statistical Methods.
• Gelman, A. et. al. (2003) Bayesian Data Analysis.
• Gabrielson, J. and Weiner, D. (2006)
  Pharmacokinetic and Pharmcodynamic Data Analysis
  Concepts and Applications

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Questions
The outcome of any serious research can only be
to make two questions grow where only one grew
before. Thorstein Veblen (1857 - 1929)